Hypoid Gear Vehicle Axle Efficiency

I. Kakavasa,∗, A. V. Olvera, D. Dinia

aTribology Group, Department of Mechanical Engineering, Imperial College, London, UK

Abstract

In this paper, a study of a hypoid gear vehicle axle is presented. Using a custom rig, load-independent losses have been accurately measured and the effect of viscosity on spin loss has beenquantified. Solution methods for the calculation of component losses are presented and combinedinto a complete thermally-coupled transient model for the estimation of axle efficiency. An anal-ysis of hypoid gear kinematics reveals a simplification, commonly adopted by other researchers,regarding the velocity of the point of contact in hypoid gears, to be in error. As a result, the cal-culation of lubrication parameters has been improved. Finally, experimental measurements arecompared to the generated simulation results for a number of operating scenarios and satisfactorycorrelation is observed.

Keywords: Elastohydrodynamics, Lubrication, Friction, Contact mechanics, Hypoid gear

1. INTRODUCTION

The increasing global energy demand com-bined with environmental concerns, as well asthe volatile value of crude oil[1] has drivengovernments and markets to pursuit higher ef-ficiency in the automotive industry through in-centives and legislation[2]. In the average pas-senger vehicle, around a third of the fuel istransformed into mechanical energy, while therest escapes the system through high exhaustgas enthalpy and cooling. Although most ofthe mechanical energy consumption is utilisedin overcoming driving resistances, a signifi-cant part is wasted in engine and transmissionlosses[3].

Transmissions and differential axles sharethe same core energy consuming components,

∗Corresponding author. Tel.: +447413535162Email address: i.n.kakavas@gmail.com (I.

Kakavas)

i.e. gear pairs and rolling element bearings.Efficient powertrain design heavily depends ona thorough understanding of the operating be-haviour of such components. Spur and heli-cal gear pairs, common in vehicle gearboxes,have been extensively studied and a number ofloss estimation methods have been proposed[4, 5, 6, 7, 8, 9]. In differential axles gear pairswith more complicated geometry, such as spi-ral bevel and hypoid gears, need to be usedto achieve transmission of power through aright angle. Hypoid gears can offer higher loadcapacity and quiet operation for a small ef-ficiency compromise, making them attractiveto axle manufacturers[10]. On the downside,the complex geometry of hypoid gears has de-layed the development of reliable tools for ef-ficiency estimation tools. Recent work on hy-poid gears involves contact models that requirea great number of input parameters, extendingto tooth cutting [11] or depending on separate

Preprint submitted to Tribology International April 19, 2016

Figure 1: Axle losses categories

software packages [12]. In this study, hypoidgear vehicle axle losses have been quantifiedthrough experimental measurement and a sim-ple axle efficiency estimation method is pro-posed.

2. EXPERIMENTAL SET-UP

The losses generated in a vehicle axle canbe categorized as load-dependent or load-independent (Fig. 1). Load-dependent losses(e.g. those due to contact friction) arise whenload is transferred in the contact of rough sur-faces in relative motion and are the result of theinteraction of lubricated rough surfaces[13].In addition to the effect of load, the rollingand sliding velocities of the surfaces, as well asthe lubrication regime, are defining parametersfor the estimation of load-dependent losses.Load-independent losses (often referred to asspin losses) are mostly associated with fluid-surface interaction. More specifically, spinlosses are the combination of oil churning andwindage loss in the immersed revolving sys-tem and are not affected by contact loads[4].Additional auxiliary losses (e.g. seal loss)are also included in the load-independent cat-egory.

The total axle losses as a function of torqueand temperature may be measured, for ex-ample, using a dynamometer together withtorquementers on input and output shafts.When no output torque is applied, the input

Figure 2: Methodology of spin loss measurement

torque may be too low for accurate measure-ment with a torquemeter. Here, a rig based onthe inertia run-down method has been utilised(Fig. 2). The rig mainly consists of an electricmotor, a clutch, an axle base and instrumenta-tion. The axle is mounted on the base and con-nected to the electric motor through the clutch.By varying the power of the electric motor, theaxle is accelerated to a desired speed. Theclutch is then disengaged and the axle is al-lowed to decelerate naturally due to losses oc-curring in the rotating axle components. Therotational speed of the axle is constantly mon-itored using an optical encoder attached to thecrown gear shaft. The captured rate of decel-eration (dω/dt) is then converted to instanta-neous axle torque loss (T ), given the rotationalinertia of the axle (I). A flywheel has been at-tached to the input flange of the rear drive unit.This increases the total effective rotational mo-ment of inertia to I = 0.035 kgm2 (referredto the high-speed shaft) prolonging the run-down period and serving to increase the time-resolution of the torque determination, partic-ularly at low speeds. The differential gears ofthe axle were fixed with structural adhesive inorder to ensure the two output shafts were syn-chronised.

In order to identify the relationship betweenviscosity and axle loss generation, lubricants

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Figure 3: Axle section

of different viscosity (but with the same chem-ical additive package and base-stock type)have been used. The temperature of the oilsump is monitored prior to run-down initia-tion. In each case, standard production oilvolume has been used allowing all four bear-ings and lower gear teeth to be sufficiently im-mersed (Fig. 3). The starting speed of the run-down was set to 2000 rev/min. Measurementswere also conducted for 1000 and 500 rev/minin order to identify potential differences due toflow behaviour. Each test was run several (3-4) times to establish repeatability and a sixth-order polynomial is produced from the averagevalues (Fig. 4).

Figure 4: Crown gear speed data during a rundown

Figure 5: Transient axle simulation model algorithm

3. AXLE SIMULATION

3.1. Transient model

As displayed in Figure 5, the basic structureof the model consists of the initial parameterinput, the theoretical component loss estima-tion, the heat transfer calculation and, finally,the axle loss/efficiency output. The transientcomponent of the algorithm can be adjustedbased on a specific drive cycle or any othertime dependent input.

Apart from the calculation of the main com-ponent losses generation, all other parts of themodel are relatively straightforward. Thereis no conventional way of calculating hypoidgear and bearing losses in the operating rangeof a passenger vehicle axle. Thus, the theo-retical basis chosen for such a computationalmodel will be described in the following sec-tions, along with the assumptions adopted forthe realisation of this model.

3.2. Driving resistance and load

The driving resistance opposing the mo-tion of a vehicle is a function of the vehicle’sspeed[14] and other design constants (Eq. 1).

Fresistance = Fdrag + Frolling + Fgradient

= 0.5ρu2vehCDA + mgCrolling + mgsinθ (1)

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Figure 6: Vehicle driving resistance

Where ρ is the density of the air, u is thespeed of the vehicle, CD is the coefficient ofdrag (typically around 0.3), A is the frontal ve-hicle area, m is the mass of the vehicle, g is thegravitational acceleration and, finally, Crolling

is the rolling friction coefficient (assumed tobe equal to 0.015).

The force (Fa) needed to increase the speedof a vehicle over a given time is equal to the ve-hicle mass times the acceleration (du/dt) mul-tiplied by a factor accounting for the rotatingcomponent inertia (km, typically between 1.08and 1.1 [15])(Eq. 2).

Fa = kmmduveh

dt(2)

The torque transmitted through an axle toovercome all opposing forces can be derivedby Equation 3, given the wheel radius (Rwheel).

Ttransmitted = (Fresistance + Fa)Rwheel (3)

The resulting axial and radial loads on theteeth of the hypoid gear pair are given, for ex-ample, by Harris[16] based on the design an-gles of the gears.

3.3. Gear loss3.3.1. Load-dependent

The load-dependent power dissipated in thegear mesh (Pmesh) is equal to the frictionalforce developed in the contact of the gear teeth(F f rictional) times the sliding velocity of thecontact (us) (Eq. 4).

Pmesh = F f rictionalus = µFloadus (4)

Transient loading and local geometry/filmthickness, although vital to durability, do notgreatly affect mean power loss. As a result,average values for the parameters in Eq. 4 areused. Given the transmitted load, an estima-tion method for the friction coefficient (µ), aswell as an analysis on hypoid gear kinematicsare necessary for the calculation of the load-dependent power loss.

Gear kinematicsIn order to analyse the lubrication of gearteeth, it is necessary to find the entrainmentvelocity (ue). This is defined[17] as the meanspeed of the contacting (gear and pinion) toothsurfaces, relative to their point of contact. Itrepresents the average speed at which the sur-faces sweep the lubricant into the region ofcontact and hence controls their lubrication. uemay be defined as:

ue ≡12

[(vg − vc) + (vp − vc)] =12

[ug + up]− uc

(5)Here, the suffices g and p indicate the actual

surfaces of the pinion and gear and u indicatesthe component in the tangent plane of the ve-locities, v. The corresponding normal veloci-ties ng, np and nc (Fig. 7) are identically equaland hence cancel from the above expression.

uc is the velocity of the point of contact it-self. In Equation 5, it is subtracted from eachof the actual surface velocities to yield the re-quired velocity relative to that of the point ofcontact.

A number of previous workers[18, 19] havemade the additional assumption that uc is zeroso that Equation 5 becomes:

ue =12

[ug + up] (6)

This assumption has probably been adopteddue to the analysis being an extension of spurgear kinematics. In spur and helical gears, the

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Figure 7: (a) Hypoid gear kinematic analysis, (b) Pro-jection on tangent plane of tooth contact, Ω

point of contact moves in a straight line normalto the tooth surfaces. Hence its velocity is al-ways normal to the tangent plane; the tangen-tial component, (uc) is indeed zero and neednot be considered. However, it is not apparentthat this is correct for hypoid gears and it iscertainly not true in the general case. Counter-examples include rolling bearings, where thepoints of contact orbit the assembly with therolling elements and hence have velocities thatare entirely tangential and cams and tappets,where the contact typically oscillates back andforth.

In addition, in the present work, we haveconsidered only the component of ue perpen-dicular to the major axis of the static area ofcontact (T), thus the component on T′ (Fig. 7).This is justifiable since entrainment parallel tothe elongated contact has little effect on the lu-brication. This is equivalent to adopting a linecontact approximation.

In order to investigate the influence of ve-locity of the point of contact more extensively,

Figure 8: (a) CAD model of gear pair, (b) Dense poly-gon mesh on hypoid gear tooth surface

a rigid tooth contact analysis method was de-veloped. First, a hypoid gear pair 3D modelwas designed (Fig. 8a) and the componentswere meshed based on high node density char-acteristics (element size = 0.5mm) (Fig. 8b).

Then, the node coordinates of the 3D partswere exported into Matlab and a program wascreated to solve for relative node distance be-tween components. Tooth contact was de-tected by applying a predefined value of prox-imity as a filter. As a result, the componentsurface velocities and the velocity of the pointof contact were calculated. A rigid surface wasused and elasticity effects were not considered.This method accounts for completely generalgeometry and does not assume theoretical linecontact or any particular solution of the law of

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Figure 9: Progression of the rotation of the gears(LEFT) and the teeth pairs contacts (RIGHT)

gearing.As the gears rotate, different nodes are

found within a predefined distance of eachother. In Figure 9, the rotation of the gear meshcan be observed along with the resulting con-tact area (red). On the first timestep displayed,two pairs of teeth are in contact. Subsequently,contact pair 1 goes out of proximity and onlycontact pair 2 continues. On the next timestep,pair 2 is closer to the end of its load sharing tra-jectory, while a third contact pair comes intocontact. In Figure 10 it is possible to followthe contact of one pair of gear teeth from startto finish, displayed along the crown gear nodesin rotation.

With the node coordinates at each timestep

Figure 10: Progression of a contact as the crown gearrotates

available, the calculation of the tooth surfaceand point of contact velocities is possible. Asseen in Figures 9 and 10, the progressing con-tact seems to be at a small angle to the tooth,as other researchers have predicted. However,it was found that the velocity of the point ofcontact (uc) is not normal to the contact planefor the hypoid gear in this application and, asa result, needs to be considered when estimat-ing the entrainment velocity. Figure 7 showsthese vectors approximately to scale for a typ-ical contact position. Thus, the correct defi-nition of the entrainment velocity is given byEquation 7:

ue =12

[ug + up] − uc (7)

The entrainment speed and sliding veloci-ties calculated in the contacts of the presenthypoid gear pair model produced an averageslide-to-roll ratio of the order of 70%. Basedon the definition of average entrainment speeddistributed along the tooth contact profile inother relevant work on hypoid gears [20], ue

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can be written as a function of ω and an effec-tive radius (Re) that depends on geometry (Eq.8). For the hypoid gear set in this project, Re

is found to be approximately 1/3 of the pitchradius of the gear:

ue = ωRe = ω0.34Rpitch (8)

Furthermore, the velocity of the point ofcontact normal to the contact plane (nc) wasfound to be equal to the corresponding gearand pinion velocities (ng,p), which confirmsthat the gears in this model are conjugate.

Friction coefficientFor the estimation of the gear friction co-

efficient, the method described by Olver andSpikes [21] has been utilised. This approachprovides a simple semi-analytical predictivetool that has the potential to be useful for com-paring designs and lubricants.

The method utilises a limited shear stressapproach, previously developed by Johnsonand others [22, 23, 24]. The friction coeffi-cient of a contact (µ f ) is given by the meanshear stress (τ) over the mean pressure in thecontact (p) (Eq. 9). For τ equal to the non-dimensional shear stress (τ∗) times the Eyringstress (τE), we define:

µ f =τ

p=τ∗τE

p(9)

The following non-dimensional values aredefined (Eq. 10):

S =γη0

τE, D0 =

η0ue

2aGe, τ∗c =

τc

τE(10)

Here S is the non-dimensional strain rateand is a function of the strain rate (γ),the dynamic viscosity (η0) and the Eyringstress. The Deborah number (D0) is a non-dimensional parameter used to characterisefluid elasticity. It is a function of dynamicviscosity, entrainment speed (ue), the lubricant

pressure-viscosity coefficient (a) and the elas-tic shear modulus (Ge). Finally, τ∗c is the non-dimensional limiting shear stress, which is thelimit that is employed when the shear rate ex-ceeds the maximum for Newtonian behaviour,and is equal to the limiting shear stress (τc)over the Eyring stress.

In order to calculate the viscosity of thelubricant inside the contact (ηr) under pres-sure, the formula provided by Roelands [25]is adopted (Eq. 11).

ηr = η0e( p′az [(1+

pp′ )

z−1]) (11)

Where η0 is the lubricant viscosity at atmo-spheric pressure and p

′

, z are constants takenas 1.9 ∗ 108 and 0.6 respectively.

Next, the mean shear stress formula dependson the values of the Deborah number and thenon-dimensional strain rate. There are two cri-teria to determine the appropriate formula.

• Is the response viscoelastic? (D0 > 1)

• Is there shear thinning? (S > sinh 1)

Knowing the answer to those questions, theshear stress can be calculated using Table 1.

S ≤ sinh 1

D0 ≤ 1 τ∗ = SD0 > 1 τ∗ = S [1 − D0(1 − e−1/D0)]

S > sinh 1

D0 ≤ 1 τ∗ = sinh−1 SD0 > 1 τ∗ ≈

ln (2S )1 − ln (1 + 2|S − 12 |e−S/2D0)

Table 1: Mean shear stress determination

However, if the shear stress exceeds its limit7

(Eq. 12):τ∗ > τ∗c (12)

Then, it is taken as the limited value (Eq. 13):

τ∗ = τ∗c = 0.06p/τE (13)

For gears, a correction needs to be appliedto the friction coefficient, to account for thecyclic variation of contact conditions, as wellas, the effect of surface roughness. The sug-gested equation to fit µ to a gear pair includesexperimental measurements of boundary fric-tion, thus, making it more reliable (Eq. 14) .

µ = µ f +µb − µ f

(1 + λ)m (14)

Where µ f is the friction coefficient as calcu-lated through the procedure described above,µb is the coefficient of boundary friction asmeasured from experiments, m is an exponentcoefficient, and λ is the lambda ratio of thecontact surfaces. Attempting to correlate theestimation of µwith previous friction measure-ments for similar operating conditions, it wasfound that the best fit occurs with µb = 0.12and m = 7.

3.3.2. Load-independentA gear immersed in lubricant will displace

oil while in operation. The energy consumedby this process is called the churning loss. Af-ter a number of experiments on a similar hy-poid gear axle, Jeon [26] proposed a formulathat best fitted his measurements of churninglosses (Eq. 15 and 16).

Tch = 0.5ρlubω2R2

pbS mCm (15)

Cm = 10(hlub

Rp)0.1786(

Vlub

R3p

)−0.2195

Re−04169Fr−0.4482(νlub

νair)−0.1777 (16)

For the calculation of the churning torque(Tch), parameters that are taken into accountinclude lubricant immersion (hlub), gear geom-etry (gear pitch radius, Rp, thickness, b, im-mersed area, S m), the viscosities of the lu-bricant (νlub) and air (νair), as well as operat-ing parameters such as contact/lubricant kine-matics represented by the Reynolds (Re) andFroude numbers (Fr) and the rotational speedof the gear (ω).

3.4. Bearing loss

The amount of loss generated by the sup-porting bearings can be calculated using theempirical method developed by SKF[27].

M = φishφrsMrr + Msl + Mseal + Mdrag (17)

As seen in Equation 17, the total frictionalmoment (M) of each bearing is given as a func-tion of rolling friction (M(rr)), sliding fric-tion (Msl), seal friction (Mseal), and oil drag(Mdrag). The effects of shear heating (φish) andstarvation (φrs) on rolling friction are also ac-counted for.

3.5. Thermal model

While friction and churning inside the axlegenerate heat and increase the temperature ofthe components, air convection on the hous-ing of the axle, occurring on a moving vehi-cle, is cooling the system down. Heat trans-fer needs to be calculated for the predictionof the loss profile of a transient system to bepossible. In reality, temperature varies acrossthe mass of the axle. However, here we as-sume that the axle is at a uniform tempera-ture identical to that of the oil supplied to thegear mesh. In order to calculate the externalconvective heat transfer coefficient, the shapeof the axle is considered to be a cylinder offinite length. Adopting these simplifications,

8

the temperature after each iteration can be cal-culated.

If the lumped mass absorbed all the gener-ated heat, then the temperature of the axle afterone timestep would become (Eq. 18):

Tt+1 = Tt +Q

meCp(18)

Here Q is the net heat generated in theaxle within the timestep, me is the equivalentlumped mass of the axle, and Cp is the heatcapacity of the equivalent mass.

Due to air flow on the axle housing, heat istransferred from the components to the air (Eq.19).

Qconvection = hAaxle∆T (19)

The heat transfer coefficient due to forcedconvection (h) can be calculated using Equa-tion 20:

h =kNu

D(20)

With Nu being the Nusselt number, D thecharacteristic length parameter of the shapewithin the flow, and k the thermal conductivityof the fluid which in this case is the environ-mental air.

For a cylinder of finite length the Nusseltnumber is given as a function of the Prandtland Reynolds numbers from the Churchill-Bernstein equation [28] (Eq. 21):

Nu = 0.3+

0.62Re0.5Pr1/3

(1 + (0.4Pr )2/3)1/4

[1 + (Re

282000)5/8]4/5 (21)

Where:

Pr =ν

at, Re =

uDν

Here u is the velocity of the flow (takenequal to the speed of the vehicle), at the ther-mal diffusivity, D the characteristic dimensionof the shape, and ν the kinematic viscosity.

Finally, the actual temperature of the axle(T ∗t+1) as a result of the effects of internal heatgeneration and forced convection within theduration of the chosen timestep is given bycombining Equations 18 and 19 into 22:

T ∗t+1 = Tenv + e−hAaxle

mCpdt(Tt+1 − Tenv) (22)

4. RESULTS

4.1. Spin loss measurementsLoad-independent losses for a hypoid gear

axle were measured using the experimental setup detailed in section 2. Initially, the axledeceleration was measured at four viscositypoints as seen in Figure 11.

These measurements were repeated forthree different starting speeds (2000, 1000 and500 rev/min) and the generated data were con-verted to axle torque and power loss. Addi-tionally, losses were estimated using the axlemodel proposed in Section 3. In Figure 12 thetorque and power losses generated in the axlewith different viscosity lubricant are presentedalong with the relevant prediction results.

Figure 11: Axle deceleration data

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Figure 12: Correlation of predicted steady-state spin torque and power loss with run-down measurements at each ofthree different starting speeds. (a) lubricant A at 30 C, (b) lubricant A at 50 C, (c) lubricant B at 30 C and (d)lubricant B at 50 C

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Figure 13: Axle input torque required for the presentpassenger vehicle over the NEDC

As seen in these graphs, the prediction cor-relates well with the spin loss measurements.However, at very low speeds, the model seemsunable to fully capture the frictional behaviourof the axle. This is probably due to the factthat the transition to mixed lubrication is notexplicitly modelled for the bearings. Never-theless, the prediction of the rest of the speedrange fits the measurements accurately andprovides a useful design tool, especially con-sidering the small contribution of loss under200 rev/min.

4.2. NEDC simulation

The production axle utilised in this studyhad previously been tested on a vehicle,thus, loss and temperature measurements wereavailable for the New European Drive Cycle(NEDC). This is a mild, mainly urban drivecycle on level terrain. Subsequently, the pre-dictive model was used to simulate the drivecycle. The axle input torque required by thepresent vehicle over the NEDC is shown inFigure 13.

In Figure 14, the correlation between thesimulation and the measured NEDC axlelosses on the vehicle is shown. For the mostpart, losses are predicted accurately and fol-low the measured trends. At the highest speed

Figure 14: Axle power loss prediction compared withon-vehicle measurements over the NEDC

however, there is up to 15% discrepancy, pos-sibly due to the lumped mass thermal modelbeing unable to account for rapid local tem-perature rises in the brief period of high speedin this drive cycle. Increasing the fidelity ofthe thermal model by mapping discrete areasof the axle could significantly improve corre-lation with measurements [29].

In Figure 15, the temperature progression ofthe lubricant sump from the probe in the testedaxle is compared to the prediction obtained us-ing the lumped mass thermal model. Again,the predicted temperature rise is within a few% with the largest discrepancy in rate of tem-perature rise at the end of the test where thespeed is highest.

With the calculated overall axle loss provid-ing good correlation to on-vehicle measure-ments, the individual component loss contri-bution can be estimated. As presented in Fig-ure 16, bearing loss (contact friction and lu-bricant churning) dominates the overall axleloss over the NEDC. During steady speed in-tervals, gear friction and churning loss amountto a very small contribution. However, whilethe vehicle is accelerating, gear friction causessignificant power losses due to the added iner-tial load. A greater proportion of gear losseswould, of course be expected for a drive cycle

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Figure 15: Axle lumped mass temperature predictioncompared with bulk oil on-vehicle measurements overthe NEDC

Figure 16: Breakdown of estimated axle loss over theNEDC for Axle lumped mass temperature predictioncompared with bulk oil on-vehicle measurements overthe NEDC

that included more severe (higher torque) con-ditions, such as high speed or inclined terrain.

5. CONCLUSIONS

A method to calculate hypoid gear axle ef-ficiency has been presented, using a transientthermally coupled model of individual compo-nent losses estimation.

A kinematic analysis on hypoid gears hasrevealed a common misconception regardingthe estimation of the entrainment speed in ahypoid gear tooth contact. A new definition

has been proposed that accounts for the veloc-ity of the point of contact and produces moreaccurate results. Furthermore, experimentaldata on axle losses has been generated utilis-ing a custom in-house rig and used, along withavailable data on the performance of a passen-ger vehicle under the NEDC, to validate themodel.

Satisfactory correlation between measure-ments and simulation has been achieved, al-lowing for the use of the proposed model withconfidence. As expected, hypoid gear axlelosses heavily depend on lubricant viscosity.Results show that component contribution tolosses varies with acceleration, with bearingsappearing as the dominant source of loss underthe NEDC. This suggests that rebalancing thefocus of component optimisation has the po-tential to introduce significant improvementsin terms of efficiency in transmission systemsbased on hypoid gears.

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NOTATION

A Vehicle frontal area

Aaxle Axle frontal area

CD Coefficient of drag

Cp Heat capacity

Crolling Coefficient of rolling friction

D0 Deborah number

Fr Froude number

Ge Elastic shear modulus

Nu Nusselt number

Pr Prandtl number

Re Reynolds number

Rpitch Pitch radius

S Non-dimensional strain rate

S m Gear immersed area

Tt Temperature of lumped mass at t

Tch Churning torque loss

Tenv Environmental temperature

Ttransmitted Transmitted torque

Vlub Lubricant volume

a Pressure-viscosity coefficient

at Thermal diffusivity

b Gear thickness

g Gravitational acceleration

h Heat transfer coefficient

hlub Lubricant immersion

km Rotating component inertia factor

me Axle equivalent lumped mass

ng,p Gear or pinion velocity normal to thecontact plane

p Mean pressure in contact

t Time

u′c Component of the velocity of thepoint of contact on the contact planeperpendicular the the line of contact

ue Entrainment velocity

u′g,p Component of the gear or piniontooth velocity on the contact planeperpendicular the the line of contact

ug,p Gear or pinion tooth velocity on thecontact plane

us Sliding speed

uveh Vehicle speed

vg,p Gear or pinion tooth velocity

Ω Contact plane

γ Strain rate

η0 Dynamic viscosity

λ Lambda ratio

µ Friction coefficient

µb Coefficient of boundary friction

ν Kinematic viscosity

ρ Density

τ Mean shear stress in contact

τ∗ Non-dimensional shear stress

τc Limiting shear stress

τ∗c Non-dimensional limiting shearstress

τE Eyring stress

φish Shear heating coefficient

φrs Starvation coefficient

ω Angular velocity

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